Question

1. find the distance between the two points in simplest radical form.

(5, 1) and (-3, -3)

Explanation:

Step1: Identify distance formula

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$.

Step2: Substitute values

Let $(x_1,y_1)=(5,1)$ and $(x_2,y_2)=(-3,-3)$. Then $x_2 - x_1=-3 - 5=-8$ and $y_2 - y_1=-3 - 1=-4$.

Step3: Calculate squares and sum

$(x_2 - x_1)^2=(-8)^2 = 64$ and $(y_2 - y_1)^2=(-4)^2 = 16$. The sum is $64 + 16=80$.

Step4: Simplify the radical

$d=\sqrt{80}=\sqrt{16\times5}=4\sqrt{5}$.

Answer:

$4\sqrt{5}$